Question: Simplify the following expression and state the condition under which the simplification is valid: $k = \dfrac{p^2 - 5p - 36}{p^2 - 16}$
First factor the expressions in the numerator and denominator. $ \dfrac{p^2 - 5p - 36}{p^2 - 16} = \dfrac{(p - 9)(p + 4)}{(p - 4)(p + 4)} $ Notice that the term $(p + 4)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(p + 4)$ gives: $k = \dfrac{p - 9}{p - 4}$ Since we divided by $(p + 4)$, $p \neq -4$. $k = \dfrac{p - 9}{p - 4}; \space p \neq -4$